Asara b’Tevet gets little respect, even in the frum community. In our hemisphere, it starts late, ends early, and is probably the least physically taxing of the fast days. Oddly, the relative ease of observance of the fast may have the perverse effect of discouraging widespread observance. Perhaps the thinking is that if it’s easy to fast for about 12 hours, the day is not really serious enough to merit much attention. However, the historical event associated with the day-- the beginning of the siege on Jerusalem -- is really quite calamitous. Little imagination is required to see why this was demonstrably the beginning of the end.
To me, Asara b’Tevet is not qualitatively much different than the eventual breach of the city walls commemorated on the 17th of Tammuz, and in fact
Zechariah 8:19 seems to treat equally the entire group of fast days commemorating the destruction.
p.s. If you’re ever in Lower Manhattan,
the Wall Street Synagogue is a good place for mincha (and really the only game in town on fast days).